BOUNDARY CONTROL LAWS FOR PARABOLIC PDES WITH VOLTERRA NONLINEARITIES—PART I: DESIGN

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolyc PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains Tn of increasing dimensions n +1 and with a domain shape in the form of a “hyper-pyramid,” 0 ≤ ζn ≤ ζn–1... ≤ ζ1 ≤ x ≤ 1. This is the first part of a three-part paper; in the second part we illustrate our design method with several examples and in the third part we provide a theoretical study of the properties of the transformation.